Flach 和 Morin 在 (Doc. Math. 23 (2018), 1425--1560) 中为一个适当的、正则的算术方案 X（即分离且有限类型 over Spec ℤ）构造了 Weil-étale 上同调 H^i_W,c (X, ℤ(n))，其中 n ∈ ℤ。在 n < 0 的情况下，他们将这种构造推广到任意的算术方案 X，从而去除了适当的和正则的假设。该构造使用了 étale 动机上同调群 H^i(X_ét, ℤ^c(n))，如 Geisser (Ann. of Math. (2) 172 (2010), 1095--1126) 所研究的，假设其在 n < 0 时的有限生成性。我们给出了一类 X，其中有限生成性已知，因此 H^i_W,c (X, ℤ(n)) 在无条件下定义。
英文摘要：
Flach and Morin constructed in (Doc. Math. 23 (2018), 1425--1560) Weil-étale cohomology $H^i_\text{W,c} (X, \mathbb{Z} (n))$ for a proper, regular arithmetic scheme $X$ (i.e. separated and of finite type over $\operatorname{Spec} \mathbb{Z}$) and $n \in \mathbb{Z}$. In the case when $n < 0$, we generalize their construction to an arbitrary arithmetic scheme $X$, thus removing the proper and regular assumption. The construction uses étale motivic cohomology groups $H^i(X_\text{ét}, \mathbb{Z}^c(n))$, as studied by Geisser (Ann. of Math. (2) 172 (2010), 1095--1126), and assumes their finite generation for $n < 0$. We give a class of X for which finite generation is known, and hence $H^i_\text{W,c} (X, \mathbb{Z} (n))$ is defined unconditionally.
In this paper we investigate linear dependence of points in Mordell-Weil groups of abelian varieties via reduction maps. In particular we try to determine the conditions for detecting linear dependence in Mordell-Weil groups via finite number of reductions.
We define a family of arithmetic zero cycles in the arithmetic Chow group of a modular curve X_0(N), for N>3 odd and squarefree, and identify the arithmetic degrees of these cycles as q-coefficients of the central derivative of a Siegel Eisenstein series of genus two. This parallels work of Kudla-Rapoport-Yang for Shimura curves.
We prove an analogue for algebraic stacks of Hermite-Minkowski's finiteness theorem from algebraic number theory, and establish a Chevalley-Weil type theorem for integral points on stacks. As an application of our results, we prove analogues of the Shafarevich conjecture for some surfaces of general type.
We define and study a Weil-étale topos for any regular, proper scheme $X$ over $\Spec(Z)$ which has some of the properties suggested by Lichtenbaum for such a topos. In particular, the cohomology with $R$-coefficients has the expected relation to $ζ(X,s)$ at $s=0$ if the Hasse-Weil L-functions $L(h^i(X_Q),s)$ have the expected meromorphic continuation and functional equation. If $\X$ has characteristic $p$ the cohomology with $Z$-coefficients also has the expected relation to $ζ(X,s)$ and our cohomology groups recover those previously studied by Lichtenbaum and Geisser.
We formulate and prove a local arithmetic Siegel--Weil formula for GSpin Rapoport--Zink spaces, which is a precise identity between the arithmetic intersection numbers of special cycles on GSpin Rapoport--Zink spaces and the derivatives of local representation densities of quadratic forms. As a first application, we prove a semi-global arithmetic Siegel--Weil formula as conjectured by Kudla, which relates the arithmetic intersection numbers of special cycles on GSpin Shimura varieties at a place of good reduction and the central derivatives of nonsingular Fourier coefficients of incoherent Sie…
We give a conjectural description of the vanishing order and leading Taylor coefficient of the Zeta function of a proper, regular arithmetic scheme $\mathcal{X}$ at any integer $n$ in terms of Weil-étale cohomology complexes. This extends work of Lichtenbaum \cite{Lichtenbaum05} and Geisser \cite{Geisser04b} for $\mathcal{X}$ of characteristic $p$, of Lichtenbaum \cite{li04} for $\mathcal{X}=\mathrm{Spec}(\mathcal{O}_F)$ and $n=0$ where $F$ is a number field, and of the second author for arbitrary $\mathcal{X}$ and $n=0$ \cite{Morin14}. We show that our conjecture is compatible with the Tamaga…
We introduce the use of $p$-descent techniques for elliptic surfaces over a perfect field of characteristic not $2$ or $3$. Under mild hypotheses, we obtain an upper bound for the rank of a non-constant elliptic surface. When $p=2$, this bound is an arithmetic refinement of a well-known geometric bound for the rank deduced from Igusa's inequality. This answers a question raised by Ulmer. We give some applications to rank bounds for elliptic surfaces over the rational numbers.
We construct natural Green forms for special cycles in orthogonal and unitary Shimura varieties, in all codimensions, and, for compact Shimura varieties of type O(p,2) and U(p,1), we show that the resulting local archimedean height pairings are related to special values of derivatives of Siegel Eisentein series. A conjecture put forward by Kudla relates these derivatives to arithmetic intersections of special cycles, and our results settle the part of his conjecture involving local archimedean heights.
A well known argument by Serre shows that there is no Weil cohomology theory with real coefficients for smooth projective varieties over $\bar{\mathbb{F}}_p$. In this note we explain why no "Weil-"cohomology theory with real coefficients can exist for arithmetic schemes over spec $\mathbb{Z}$, even for spectra of number rings.
We study the arithmetic of the twist of the Fermat quartic defined by $X^4 + Y^4 + Z^4 = 0$ which has no $\mathbb{Q}$-rational point. We calculate the Mordell--Weil group of the Jacobian variety explicilty. We show that the degree $0$ part of the Picard group is a free $\mathbb{Z}/2\mathbb{Z}$-module of rank $2$, whereas the Mordell--Weil group is a free $\mathbb{Z}/2\mathbb{Z}$-module of rank $3$. Thus the relative Brauer group is non-trivial. We also show that this quartic violates the local-global property for linear determinantal representations.
We would like to construct a new Grothendieck topology for arithmetic schemes, whose cohomology groups associated with motivic complexes of sheaves are finitely generated and whose Euler characteristics are related to special values of zeta-functions. In this paper we construct this topology for rings of algebraic integers and show that the cohomology of the constant sheaf Z is related to the behavior of zeta-functions at s = 0.
We study the arithmetic of abelian varieties over $K=k(t)$ where $k$ is an arbitrary field. The main result relates Mordell-Weil groups of certain Jacobians over $K$ to homomorphisms of other Jacobians over $k$. Our methods also yield completely explicit points on elliptic curves with unbounded rank over $\Fpbar(t)$ and a new construction of elliptic curves with moderately high rank over $\C(t)$.
Lichtenbaum conjectured the existence of a Weil-étale cohomology in order to describe the vanishing order and the special value of the Zeta function of an arithmetic scheme $\mathcal{X}$ at $s=0$ in terms of Euler-Poincaré characteristics. Assuming the (conjectured) finite generation of some étale motivic cohomology groups we construct such a cohomology theory for regular schemes proper over $\mathrm{Spec}(\mathbb{Z})$. In particular, we obtain (unconditionally) the right Weil-étale cohomology for geometrically cellular schemes over number rings. We state a conjecture expressing the vanishing …
One of the driving motivations to develop $\F_1$-geometry is the hope to translate Weil's proof of the Riemann hypothesis from positive characteristics to number fields, which might result in a proof of the classical Riemann hypothesis. The underlying idea is that the spectrum of $\Z$ should find an interpretation as a curve over $\F_1$, which has a completion $\bar{\Spec\Z}$ analogous to a curve over a finite field. The hope is that intersection theory for divisors on the arithmetic surface $\bar{\Spec\Z} \times \bar{\Spec\Z}$ will allow to mimic Weil's proof. It turns out that it is possible…
We express some basic properties of Deninger's conjectural dynamical system in terms of morphisms of topoi. Then we show that the current definition of the Weil-étale topos satisfies these properties. In particular, the flow, the closed orbits, the fixed points of the flow and the foliation in characteristic $p$ are well defined on the Weil-étale topos. This analogy extends to arithmetic schemes. Over a prime number $p$ and over the archimedean place of $\mathbb{Q}$, we define a morphism from a topos associated to Deninger's dynamical system to the Weil-étale topos. This morphism is compatible…
The statement of the Riemann hypothesis makes sense for all global fields, not just the rational numbers. For function fields, it has a natural restatement in terms of the associated curve. Weil's work on the Riemann hypothesis for curves over finite fields led him to state his famous "Weil conjectures", which drove much of the progress in algebraic and arithmetic geometry in the following decades. In this article, I describe Weil's work and some of the ensuing progress: Weil cohomology (etale, crystalline); Grothendieck's standard conjectures; motives; Deligne's proof; Hasse-Weil zeta functio…
In this paper we investigate a local to global principle for Galois cohomology of number fields with coefficients in the Tate module of an abelian variety. In \cite{bk13} G. Banaszak and the author obtained the sufficient condition for the validity of the local to global principle for {é}tale $K$-theory of a curve . This condition in fact has been established by means of an analysis of the corresponding problem in the Galois cohomology. We show that in some cases this result is the best possible i.e if this condition does not hold we obtain counterexamples. We also give some examples of curves…
We construct a cohomology theory with compact support H^i_c(X_ar,Z(n))$ for separated schemes of finite type over a finite field, which should play a role analog to Lichtenbaum's Weil-etale cohomology groups for smooth and projective schemes. In particular, if Tate's conjecture holds and rational and numerical equivalence agree up to torsion, then the groups H^i_c(X_ar,Z(n)) are finitely generated, form an integral version of l-adic cohomology with compact support, and admit a formula for the special values of the zeta-function of X.
Lichtenbaum has conjectured the existence of a Grothendieck topology for an arithmetic scheme $X$ such that the Euler characteristic of the cohomology groups of the constant sheaf $\mathbb{Z}$ with compact support at infinity gives, up to sign, the leading term of the zeta-function $ζ_X(s)$ at $s=0$. In this paper we consider the category of sheaves $\bar{X}_L$ on this conjectural site for $X=Spec(\mathcal{O}_F)$ the spectrum of a number ring. We show that $\bar{X}_L$ has, under natural topological assumptions, a well defined fundamental group whose abelianization is isomorphic, as a topologic…
In this paper we construct a projective action of certain arithmetic group on the derived category of coherent sheaves on an abelian scheme $A$, which is analogous to Weil representation of the symplectic group. More precisely, the arithmetic group in question is a congruence subgroup in the group of "symplectic" automorphisms of $A\times\hat{A}$ where $\hat{A}$ is the dual abelian scheme. The "projectivity" of this action refers to shifts in the derived category and tensorings with line bundles pulled from the base. In particular, if $A$ is an abelian scheme over $S$ equipped with an ample li…
The aim of this paper is to compute the Frobenius structures of some cohomological operators of arithmetic $\ms{D}$-modules. To do this, we calculate explicitly an isomorphism between canonical sheaves defined abstractly. Using this calculation, we establish the relative Poincaré duality in the style of SGA4. As another application, we compare the push-forward as arithmetic $\ms{D}$-modules and the rigid cohomologies taking Frobenius into account. These theorems will lead us to an analog of "Weil II" and a product formula for $p$-adic epsilon factors.
For singular metrics, there is no Quillen metric formalism on cohomology determinant. In this paper, we develop an admissible theory, with which the arithmetic Deligne-Riemann-Roch isometry can be established for singular metrics. As an application, we first study Weil-Petersson metrics and Takhtajan-Zograf metrics on moduli spaces of punctured Riemann surfaces, and then give a more geometric interpretation of our determinant metrics in terms of Selberg zeta functions. We end this paper by proposing an arithmetic factorization for Weil-Petersson metrics, cuspidal metrics and Selberg zeta funct…
For an arithmetic surface X and a Weil divisor $D$, there are natural arithmetic cohomology groups $H_{\mathrm{ar}}^i(X, \mathcal O_X (D))$ $(i=0,1,2)$. Using ind-pro topology on adelic space $\mathbb A_{X, 012}^{\mathrm{ar}}$, we show that $H_{\mathrm{ar}}^0(X, \mathcal O_X (D))$ is discrete, $H_{\mathrm{ar}}^1(X, \mathcal O_X (D))$ is finite, and $H_{\mathrm{ar}}^2(X, \mathcal O_X (D))$ is compact. Moreover, we prove that all possible summations of canonical subspaces $\mathbb A_{X,i}^{\mathrm{ar}}(D),$ $\mathbb A_{X, kl}^{\mathrm{ar}}(D)$ $(i,k,l=0,1,2)$ are closed in $\mathbb A_{X,012}^{\m…
Let $X$ be a smooth projective variety over $ \overline{\mathbb Q}$, and $f:X -rightarrow X$ be a dominant rational map. Let $δ_{f}$ be the first dynamical degree of $f$ and $h_{X}:X( \overline{\mathbb Q})\to [1,\infty)$ be a Weil height function on $X$ associated with an ample divisor on $X$. We prove several inequalities which give upper bounds of the sequence $(h_X (f^n(P)))_{n\geq0}$ where $P$ is a point of $X( \overline{\mathbb Q})$ whose forward orbit by $f$ is well-defined. As a corollary, we prove that the upper arithmetic degree is less than or equal to the first dynamical degree; $ \…
Let $p$ be a prime, let $r$ and $q$ be powers of $p$, and let $a$ and $b$ be relatively prime integers not divisible by $p$. Let $C/\mathbb F_{r}(t)$ be the superelliptic curve with affine equation $y^b+x^a=t^q-t$. Let $J$ be the Jacobian of $C$. By work of Pries--Ulmer, $J$ satisfies the Birch and Swinnerton-Dyer conjecture (BSD). Generalizing work of Griffon--Ulmer, we compute the $L$-function of $J$ in terms of certain Gauss sums. In addition, we estimate several arithmetic invariants of $J$ appearing in BSD, including the rank of the Mordell--Weil group $J(\mathbb F_{r}(t))$, the Faltings …
In this paper, we define a certain Hodge-theoretic structure for an arbitrary variety X over the complex number field by using the theory of mixed Hodge module due to Morihiko Saito. We call it an arithmetic Hodge structure of X. It is shown that extension groups of arithmetic Hodge structure do not vanish even for degree $\geq2$. Moreover, we define higher Abel-Jacobi maps from Bloch's higher Chow groups of X to these extension groups. These maps essentially involve the classical Abel-Jacobi maps by Weil and Griffiths, and Mumford's infinitesimal invariants of 0-cycles on surfaces.
Archimedean cohomology provides a cohomological interpretation for the calculation of the local L-factors at archimedean places as zeta regularized determinant of a log of Frobenius. In this paper we investigate further the properties of the Lefschetz and log of monodromy operators on this cohomology. We use the Connes-Kreimer formalism of renormalization to obtain a fuchsian connection whose residue is the log of the monodromy. We also present a dictionary of analogies between the geometry of a tubular neighborhood of the ``fiber at arithmetic infinity'' of an arithmetic variety and the compl…
We study arithmetical and geometrical properties of {\it maximal curves}, that is, curves defined over the finite field $\mathbb F_{q^2}$ whose number of $\mathbb F_{q^2}$-rational points reachs the Hasse-Weil upper bound. Under a hypothesis on non-gaps at rational points we prove that maximal curves are $\mathbb F_{q^2}$-isomorphic to $y^q+y=x^m$ for some $m\in \mathbb Z^+$.
